METRIC SPACES EXAM WITH COMPLETE
SOLUTION
d is called a metric on X and (X,d) is called a metric space if - ANSWER
Suppose d: X × X→R and that for all x, y, z ∈ X:
1) d(x,y) ≥ 0 and d(x,y) > 0 when x does not equal y and d(x,x) = 0
2.) d(x,y) = d(y,x)
3.) d(x,z) ≤ d(x,y) + d(x,z)
every metric space is automatically - ANSWER a pseudometric space
Suppose (X,d) is a pseuodmetric space, that x₀ ∈ X and ε > 0. Then B(x₀, ε) =
{x∈X: d(x,x₀) < ε} is called - ANSWER the ball of radius ε with center at x₀
If there exists an ε > 0 such that B(x₀, ε) = {x₀}, then we say that x₀ is -
ANSWER an isolated point in (X,d)
Suppose (X,d) is a pseudometric space and O⊆X. We say that O is open in
(X,d) if - ANSWER for each x∈O there is an ε > 0 such that B(x, ε)⊆O
A set O⊆X is open iff - ANSWER O is a union of a collection of balls
a) Each ball B(x, ε) is
b) A point x₀ in a pseudometric space (X,d) is isolated iff - ANSWER open in
(X,d)
{x₀} is an open set
Suppose (X,d) is a pseudometric space. The topology Td generated by d is the -
ANSWER collection of all open sets in (X,d). In other words, Td = {O: O is
open in (X,d)} = {O: O is a union of balls}
, Let Td be the topology in (X,d). Then - ANSWER 1)∅, X ∈T
2) if Oₙ∈T for each n ∈ N, then ∪(n∈N) Oₙ ∈ T
3) if O₁, . . . , Oₙ ∈T then O₁∩...∩Oₙ ∈ T
2 says that the collection T is closed under unions and 3 says it is closed under
finite intersections
Suppose d and d' are two pseudo-metrics (or metrics) on a set X. We say that d
and d' are equivalent (written d~d') - ANSWER Td=Td', that is if d and d'
generate the same collection of open sets
A subset I of R is convex if whenever - ANSWER x≤y≤z and x, z ∈ I, then y ∈
I.
A convex subset of R is called an - ANSWER interval
I⊆R is an interval iff - ANSWER I has one of the following forms (where a<b)
(-∞,∞), (-∞,a), (-∞,a], [a,∞), (a,∞), (a,b), [a,b), (a,b], [a,b], {a}, ∅
Suppose I is a collection of intervals in R. If i ∩ j is nonempty for all i, j ∈ I,
then - ANSWER ∪I is an interval.
In particular, if ∩I is nonempty, then ∪I is an interval.
Suppose O⊆R. O is open in R iff - ANSWER O is the union of a countable
collection of pairwise disjoint open intervals
Suppose F⊆X. We say that F is closed in (X,d) if - ANSWER X - F is open
F is closed iff - ANSWER X - F is open
for all x∈X - F there is an ε > 0 for which B(x, ε) ⊆ X-F
for all x∈X-F, there is an ε > 0 for which B(x, ε) ∩ F =∅
SOLUTION
d is called a metric on X and (X,d) is called a metric space if - ANSWER
Suppose d: X × X→R and that for all x, y, z ∈ X:
1) d(x,y) ≥ 0 and d(x,y) > 0 when x does not equal y and d(x,x) = 0
2.) d(x,y) = d(y,x)
3.) d(x,z) ≤ d(x,y) + d(x,z)
every metric space is automatically - ANSWER a pseudometric space
Suppose (X,d) is a pseuodmetric space, that x₀ ∈ X and ε > 0. Then B(x₀, ε) =
{x∈X: d(x,x₀) < ε} is called - ANSWER the ball of radius ε with center at x₀
If there exists an ε > 0 such that B(x₀, ε) = {x₀}, then we say that x₀ is -
ANSWER an isolated point in (X,d)
Suppose (X,d) is a pseudometric space and O⊆X. We say that O is open in
(X,d) if - ANSWER for each x∈O there is an ε > 0 such that B(x, ε)⊆O
A set O⊆X is open iff - ANSWER O is a union of a collection of balls
a) Each ball B(x, ε) is
b) A point x₀ in a pseudometric space (X,d) is isolated iff - ANSWER open in
(X,d)
{x₀} is an open set
Suppose (X,d) is a pseudometric space. The topology Td generated by d is the -
ANSWER collection of all open sets in (X,d). In other words, Td = {O: O is
open in (X,d)} = {O: O is a union of balls}
, Let Td be the topology in (X,d). Then - ANSWER 1)∅, X ∈T
2) if Oₙ∈T for each n ∈ N, then ∪(n∈N) Oₙ ∈ T
3) if O₁, . . . , Oₙ ∈T then O₁∩...∩Oₙ ∈ T
2 says that the collection T is closed under unions and 3 says it is closed under
finite intersections
Suppose d and d' are two pseudo-metrics (or metrics) on a set X. We say that d
and d' are equivalent (written d~d') - ANSWER Td=Td', that is if d and d'
generate the same collection of open sets
A subset I of R is convex if whenever - ANSWER x≤y≤z and x, z ∈ I, then y ∈
I.
A convex subset of R is called an - ANSWER interval
I⊆R is an interval iff - ANSWER I has one of the following forms (where a<b)
(-∞,∞), (-∞,a), (-∞,a], [a,∞), (a,∞), (a,b), [a,b), (a,b], [a,b], {a}, ∅
Suppose I is a collection of intervals in R. If i ∩ j is nonempty for all i, j ∈ I,
then - ANSWER ∪I is an interval.
In particular, if ∩I is nonempty, then ∪I is an interval.
Suppose O⊆R. O is open in R iff - ANSWER O is the union of a countable
collection of pairwise disjoint open intervals
Suppose F⊆X. We say that F is closed in (X,d) if - ANSWER X - F is open
F is closed iff - ANSWER X - F is open
for all x∈X - F there is an ε > 0 for which B(x, ε) ⊆ X-F
for all x∈X-F, there is an ε > 0 for which B(x, ε) ∩ F =∅
